Riemannian metric

Definition

A Riemannian metric is a smoothly varying positive definite inner product on the tangent spaces of a differentiable manifold, allowing one to measure distances and angles on that manifold. This concept forms the backbone of Riemannian geometry, enabling the exploration of curves, surfaces, and their properties in various contexts, from smooth manifolds to curvature concepts and beyond.

5 Must Know Facts For Your Next Test

1. A Riemannian metric allows the definition of the length of curves and the angle between tangent vectors at each point on the manifold.
2. The concept of geodesics is directly tied to the Riemannian metric; they can be thought of as the 'straightest' possible paths between points in curved spaces.
3. Riemannian metrics can be used to induce metrics on submanifolds, meaning you can carry over the measurement structure from a larger manifold to a smaller one.
4. The completeness of a Riemannian manifold is determined by whether all Cauchy sequences converge within that manifold, linking back to its metric properties.
5. Different choices of Riemannian metrics on the same manifold can lead to vastly different geometric properties and behaviors.

Review Questions

• How does the definition of a Riemannian metric influence the understanding of geodesics within a manifold?
  • The Riemannian metric provides a way to measure lengths and angles, which is crucial for defining geodesics as curves that locally minimize distance. Geodesics are determined by the behavior of tangent vectors under this metric; specifically, they are characterized by having zero acceleration with respect to the connection induced by the metric. Therefore, understanding how distances are calculated through the Riemannian metric helps clarify what it means for a path to be geodesic.
• Discuss how Riemannian metrics facilitate comparisons between different manifolds in terms of their curvature properties.
  • Riemannian metrics enable comparisons between manifolds by providing a framework for analyzing their curvature properties. Metrics allow for the application of comparison theorems, such as the Rauch comparison theorem, which uses curvature information to draw conclusions about geodesic behavior and volume growth in various settings. This comparison is essential for understanding how different manifolds behave geometrically based on their respective metrics.
• Evaluate how altering a Riemannian metric affects global properties like completeness and curvature within a given manifold.
  • Altering a Riemannian metric can significantly impact global properties such as completeness and curvature. For instance, modifying the metric could change whether all Cauchy sequences converge within the manifold, thus affecting its completeness. Additionally, different metrics can yield different curvature measurements; for example, a flat metric versus a positively curved one will yield contrasting geometric behaviors. Such changes underscore the importance of the chosen Riemannian metric in determining not just local but also global geometric characteristics of manifolds.

Related terms

Metric tensor: A symmetric, positive definite bilinear form that defines the Riemannian metric on a manifold, providing a way to calculate lengths and angles.

Geodesics:

Curves that locally minimize distances between points on a manifold, characterized by having zero acceleration with respect to the Riemannian metric.

Curvature:

A measure of how much a geometric object deviates from being flat, which can be understood through various forms like sectional curvature and Ricci curvature in Riemannian geometry.